(recall that the second equality follows from [1]}), by using similar arguments previously used in [1]}.
In particular, (REF ) and (REF ) imply that the configuration \((A,u)\in \widetilde{\mathcal {C}}\)  is a minimizer of \(\widetilde{\mathcal {F}}\)  in \(\widetilde{\mathcal {C}}.\)  In Theorem REF  we establish the uniform density estimates for the jump set \(S_u^A\)  of \(u\)  along cracks for a minimizer \((A,u)\)  of \(\widetilde{\mathcal {F}}.\)  In particular, \(S_u^A\)  is then essentially closed, and using this fact in Proposition REF  we construct a configuration \((A^{\prime },u^{\prime })\in \mathcal {C},\)  which minimizes boths \(\widetilde{\mathcal {F}}\)  and \(\mathcal {F},\)  starting from a minimizer \((A,u)\)  of \(\widetilde{\mathcal {F}}\)  in \(\widetilde{\mathcal {C}}.\)  Moreover, \((A^{\prime },u^{\prime })\)  solves both (REF ) and (REF ) and satisfies the properties stated in Theorem REF . Theorem REF  is then a direct consequence of Proposition REF , comparison arguments, the isoperimetric inequality in \(\mathbb {R}^2\) , and the equivalence of the constrained minimum problems and the unconstrained penalized minimum problem related to the energies \(\mathcal {F}^\lambda \)  and \(\widetilde{\mathcal {F}}^\lambda \) .
